Gumbel fluctuations for cover times in the discrete torus

2012/02/01 by David Belius, Belius, David · 1 citation
Mathematics · #60D05 #60G50 #60G55 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:60D05 #msc:60G50 #msc:60G55

paper · pdf · doi:10.48550/arxiv.1202.0190

41 pages. Accepted for publication in Probability Theory and Related Fields

arxiv created 2012/11/07 · arxiv updated 2012/11/08

Abstract

This work proves that the fluctuations of the cover time of simple random walk in the discrete torus of dimension at least three with large side-length are governed by the Gumbel extreme value distribution. This result was conjectured for example in the book by Aldous & Fill. We also derive some corollaries which qualitatively describe "how" covering happens. In addition, we develop a new and stronger coupling of the model of random interlacements, introduced by Sznitman, and random walk in the torus. This coupling is used to prove the cover time result and is also of independent interest.

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